Mastery Check Solutions

Show What You Know

Clothing Limited has a budget of $12,000 to restock shirts, pants, and shoes in the store. There is space for a total of 300 items. Shirts cost $35 to purchase, pants are $25, and shoes are $50. Based on the current inventory, Clothing Limited needs the combined number of pants and shoes to equal the total number of shirts purchased.

Define your variables and write a system of equations.

x : shirts
y : pants
z : shoes

$A : x + y + z = 300 B : 35 x + 25 y + 50 z = 12000 C : x = y + z$

Solve the system you created in part A.

$x = y + z x - y - z = 0 + x + y + z = 300 2 x = 300 x = 150$

Note

Rewrite C so that it is equal to zero.

A + C to solve for x.

$\begin{array}{rcl} 35 (150) + 25 y + 50 z & = & 12000 \\ 5250 + 25 y + 50 z & = & 12000 \\ 25 y + 50 z & = & 6750 \end{array}$

Note

Substitute x = 150, into B.

$y + z = x \begin{array}{ccl} (y + z = 150) (- 25) & \Rightarrow & - 25 y - 25 z = - 3750 \\ + 25 y + 50 z = 6750 \\ 25 z = 3000 \\ z = 120 \end{array}$

Note

B + – 25C

$\begin{array}{rcl} x + y + z & = & 300 \\ 150 + y + 120 & = & 300 \\ 270 + y & = & 300 \\ y & = & 30 \end{array}$

There are 150 shirts, 30 pairs of pants, and 120 pairs of shoes.

Note

Solve for y.

What if the equations were written as these inequalities:

$x + y + z \leq 300$

$35 x + 25 y + 50 z \leq 12000$

$x \leq y + z$

Would 150 shirts, 110 pairs of pants, and 40 pairs of shoes be a solution? How would this impact the budget? Show your work and explain your thinking.

$150 + 110 + 40 = 300 ✓ 35 (150) + 25 (110) + 50 (40) = 10, 000 ✓ 150 = 110 + 40 ✓$

Clothing Limited will spend $10,000 out of their $12,000 budget. This combination of items can be purchased and be $2,000 under budget.

Note

Q: What are the totals in the problem?
A: $12,000 and 300 items
This system can be solved in other ways, but the solution will remain the same. Stating your Plan to solve will be a helpful way to begin part B.
Make sure to show your work and explain your thinking.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Determine if given values represent a solution to a system of equations with three variables.
Solve a system of linear equations with three variables.
Write a system of linear equations with three variables.
Evaluate word problems for linear equations with three variables.